Nuprl Lemma : in-hull-leftmost
∀g:OrientedPlane. ∀xs:{xs:Point List| geo-general-position(g;xs)} .
(2 < ||xs||
⇒ (∀i,j:ℕ||xs||.
((¬(i = j ∈ ℤ))
⇒ ij ∈ Hull(xs)
⇒ (∃x:{k:ℕ||xs||| (¬(k = i ∈ ℤ)) ∧ (¬(k = j ∈ ℤ))}
∀y:{k:ℕ||xs||| (¬(k = i ∈ ℤ)) ∧ (¬(k = j ∈ ℤ))} . ((¬(x = y ∈ ℤ)) ⇒ (↑x L iy))))))
Proof
Definitions occuring in Statement :
in-hull: ij ∈ Hull(xs),
left-test: i L jk,
geo-general-position: geo-general-position(g;xs),
oriented-plane: OrientedPlane,
geo-point: Point,
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
guard: {T},
listp: A List+,
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
band: p ∧b q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
upto: upto(n),
from-upto: [n, m),
lt_int: i <z j,
top: Top,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
cons: [a / b],
nat_plus: ℕ+,
nat: ℕ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
ge: i ≥ j ,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
sq_stable: SqStable(P),
comparison: comparison(T),
hull-cmp: hull-cmp(g;xs;i;j),
true: True
Lemmas referenced :
hull-cmp_wf,
in-hull_wf,
istype-int,
set_subtype_base,
lelt_wf,
length_wf,
geo-point_wf,
int_subtype_base,
istype-void,
int_seg_wf,
istype-less_than,
list_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
oriented-plane-subtype,
subtype_rel_transitivity,
oriented-plane_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-general-position_wf,
filter_type,
bnot_wf,
eq_int_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bool_cases_sqequal,
eqff_to_assert,
assert-bnot,
neg_assert_of_eq_int,
bfalse_wf,
upto_wf,
subtype_rel_list,
assert_wf,
not_wf,
equal-wf-base,
istype-assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int,
assert_of_band,
list-cases,
length_of_nil_lemma,
filter_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
upto_decomp2,
add_nat_plus,
add_nat_wf,
length_wf_nat,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
nat_plus_properties,
int_seg_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
subtract_wf,
non_neg_length,
itermSubtract_wf,
int_term_value_subtract_lemma,
filter_cons_lemma,
map_cons_lemma,
filter_wf5,
map_wf,
l_member_wf,
ifthenelse_wf,
cons_wf,
comparison-max_wf,
set-valueall-type,
int-valueall-type,
left-test_wf,
sq_stable__assert,
l_all_iff,
le_wf,
member_filter,
and_wf,
equal_wf,
member_upto2,
subtype_rel_set,
nat_wf,
int_seg_subtype_nat,
istype-false,
l_member-settype
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
promote_hyp,
independent_isectElimination,
universeIsType,
sqequalRule,
functionIsType,
equalityIstype,
applyEquality,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
because_Cache,
setElimination,
rename,
sqequalBase,
equalitySymmetry,
inhabitedIsType,
setIsType,
instantiate,
dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
cumulativity,
equalityTransitivity,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
voidElimination,
setEquality,
productEquality,
isect_memberEquality_alt,
independent_pairFormation,
baseApply,
closedConclusion,
baseClosed,
productIsType,
imageElimination,
hypothesis_subsumption,
addEquality,
approximateComputation,
applyLambdaEquality,
pointwiseFunctionality,
int_eqEquality,
equalityElimination,
imageMemberEquality
Latex:
\mforall{}g:OrientedPlane. \mforall{}xs:\{xs:Point List| geo-general-position(g;xs)\} .
(2 < ||xs||
{}\mRightarrow{} (\mforall{}i,j:\mBbbN{}||xs||.
((\mneg{}(i = j))
{}\mRightarrow{} ij \mmember{} Hull(xs)
{}\mRightarrow{} (\mexists{}x:\{k:\mBbbN{}||xs||| (\mneg{}(k = i)) \mwedge{} (\mneg{}(k = j))\}
\mforall{}y:\{k:\mBbbN{}||xs||| (\mneg{}(k = i)) \mwedge{} (\mneg{}(k = j))\} . ((\mneg{}(x = y)) {}\mRightarrow{} (\muparrow{}x L iy))))))
Date html generated:
2019_10_16-PM-01_40_38
Last ObjectModification:
2018_12_21-PM-11_02_21
Theory : euclidean!plane!geometry
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